# Properties

 Label 980.1.p.c Level $980$ Weight $1$ Character orbit 980.p Analytic conductor $0.489$ Analytic rank $0$ Dimension $8$ Projective image $D_{4}$ CM discriminant -4 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 980.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.489083712380$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.137200.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{6} q^{8} + \zeta_{24}^{4} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{6} q^{8} + \zeta_{24}^{4} q^{9} -\zeta_{24}^{5} q^{10} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{13} -\zeta_{24}^{4} q^{16} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{2} q^{18} -\zeta_{24}^{3} q^{20} -\zeta_{24}^{2} q^{25} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{26} -\zeta_{24}^{2} q^{32} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{34} + q^{36} + 2 \zeta_{24}^{10} q^{37} -\zeta_{24} q^{40} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{41} -\zeta_{24}^{11} q^{45} - q^{50} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{52} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{61} - q^{64} + ( \zeta_{24}^{4} - \zeta_{24}^{10} ) q^{65} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{68} -\zeta_{24}^{10} q^{72} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{73} + 2 \zeta_{24}^{8} q^{74} + \zeta_{24}^{11} q^{80} + \zeta_{24}^{8} q^{81} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{82} + ( -1 + \zeta_{24}^{6} ) q^{85} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{89} -\zeta_{24}^{9} q^{90} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 4q^{9} + O(q^{10})$$ $$8q + 4q^{4} + 4q^{9} - 4q^{16} + 8q^{36} - 8q^{50} - 8q^{64} + 4q^{65} - 8q^{74} - 4q^{81} - 8q^{85} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-\zeta_{24}^{4}$$ $$-1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.965926 + 0.258819i 0 0 1.00000i 0.500000 0.866025i 0.965926 + 0.258819i
79.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.965926 0.258819i 0 0 1.00000i 0.500000 0.866025i −0.965926 0.258819i
79.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.258819 0.965926i 0 0 1.00000i 0.500000 0.866025i 0.258819 0.965926i
79.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.258819 + 0.965926i 0 0 1.00000i 0.500000 0.866025i −0.258819 + 0.965926i
459.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.965926 0.258819i 0 0 1.00000i 0.500000 + 0.866025i 0.965926 0.258819i
459.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.965926 + 0.258819i 0 0 1.00000i 0.500000 + 0.866025i −0.965926 + 0.258819i
459.3 0.866025 0.500000i 0 0.500000 0.866025i −0.258819 + 0.965926i 0 0 1.00000i 0.500000 + 0.866025i 0.258819 + 0.965926i
459.4 0.866025 0.500000i 0 0.500000 0.866025i 0.258819 0.965926i 0 0 1.00000i 0.500000 + 0.866025i −0.258819 0.965926i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 459.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
140.c even 2 1 inner
140.p odd 6 1 inner
140.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.p.c 8
4.b odd 2 1 CM 980.1.p.c 8
5.b even 2 1 inner 980.1.p.c 8
7.b odd 2 1 inner 980.1.p.c 8
7.c even 3 1 980.1.f.e 4
7.c even 3 1 inner 980.1.p.c 8
7.d odd 6 1 980.1.f.e 4
7.d odd 6 1 inner 980.1.p.c 8
20.d odd 2 1 inner 980.1.p.c 8
28.d even 2 1 inner 980.1.p.c 8
28.f even 6 1 980.1.f.e 4
28.f even 6 1 inner 980.1.p.c 8
28.g odd 6 1 980.1.f.e 4
28.g odd 6 1 inner 980.1.p.c 8
35.c odd 2 1 inner 980.1.p.c 8
35.i odd 6 1 980.1.f.e 4
35.i odd 6 1 inner 980.1.p.c 8
35.j even 6 1 980.1.f.e 4
35.j even 6 1 inner 980.1.p.c 8
140.c even 2 1 inner 980.1.p.c 8
140.p odd 6 1 980.1.f.e 4
140.p odd 6 1 inner 980.1.p.c 8
140.s even 6 1 980.1.f.e 4
140.s even 6 1 inner 980.1.p.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.f.e 4 7.c even 3 1
980.1.f.e 4 7.d odd 6 1
980.1.f.e 4 28.f even 6 1
980.1.f.e 4 28.g odd 6 1
980.1.f.e 4 35.i odd 6 1
980.1.f.e 4 35.j even 6 1
980.1.f.e 4 140.p odd 6 1
980.1.f.e 4 140.s even 6 1
980.1.p.c 8 1.a even 1 1 trivial
980.1.p.c 8 4.b odd 2 1 CM
980.1.p.c 8 5.b even 2 1 inner
980.1.p.c 8 7.b odd 2 1 inner
980.1.p.c 8 7.c even 3 1 inner
980.1.p.c 8 7.d odd 6 1 inner
980.1.p.c 8 20.d odd 2 1 inner
980.1.p.c 8 28.d even 2 1 inner
980.1.p.c 8 28.f even 6 1 inner
980.1.p.c 8 28.g odd 6 1 inner
980.1.p.c 8 35.c odd 2 1 inner
980.1.p.c 8 35.i odd 6 1 inner
980.1.p.c 8 35.j even 6 1 inner
980.1.p.c 8 140.c even 2 1 inner
980.1.p.c 8 140.p odd 6 1 inner
980.1.p.c 8 140.s even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{1}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$1 - T^{4} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$( 2 + T^{2} )^{4}$$
$17$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$41$ $$( -2 + T^{2} )^{4}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$97$ $$( 2 + T^{2} )^{4}$$